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24 Pages·2017·17.19 MB·English

Delft University of Technology

New insights into the short pitch corrugation development enigma based on 3D-FE
dynamic vehicle-track coupled modelling in frictional rolling contact
Li, Shaoguang; Li, Zili; Nunez, Alfredo; Dollevoet, Rolf
DOI
10.3390/app7080807
Publication date
2017
Document Version
Final published version
Published in
Applied Sciences
Citation (APA)
Li, S., Li, Z., Nunez, A., & Dollevoet, R. (2017). New insights into the short pitch corrugation development
enigma based on 3D-FE dynamic vehicle-track coupled modelling in frictional rolling contact. Applied
Sciences, 7(8), [807]. https://doi.org/10.3390/app7080807
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applied
sciences
Article
New Insights into the Short Pitch Corrugation
Enigma Based on 3D-FE Coupled Dynamic
Vehicle-Track Modeling of Frictional Rolling Contact
ShaoguangLi,ZiliLi*,AlfredoNúñez ID andRolfDollevoet
SectionofRailwayEngineering,FacultyofCivilEngineeringandGeosciences,DelftUniversityofTechnology,
Stevinweg1,2628CNDelft,TheNetherlands;[email protected](S.L.);[email protected](A.N.);
[email protected](R.D.)
* Correspondence:[email protected];Tel.:[phone]
Received:13June2017;Accepted:31July2017;Published:7August2017
Abstract: Athree-dimensional(3D)finiteelement(FE)dynamicfrictionalrollingcontactmodelis
presentedforthestudyofshortpitchcorrugationthatconsidersdirectandinstantaneouscoupling
betweenthecontactmechanicsandthestructuraldynamicsinavehicle-tracksystem. Inthisstudy,
weexaminethesystemresponsesintermsofvibrationmodes,contactforcesandtheresultingwear
withsmoothrailandcorrugatedrailwithprogressivelyincreasingamplitudetoinfertheconditions
forconsistentcorrugationinitiationandgrowth. Wearisassumedtobethedamagemechanism,and
shortpitchcorrugationismodeledusingwavelengthsfromfieldobservationsofaDutchrailway.
Thecontributionofthispaperisaglobalperspectiveoftheconsistencyconditionsthatgovernthe
evolutionofshortpitchcorrugation. Themaininsightsareasfollows: (1)thelongitudinalvibration
modesareprobablydominantforshortpitchcorrugationinitiation;(2)duringshortpitchcorrugation
evolution,theinteractionandconsistencybetweenlongitudinalandverticalmodesshoulddetermine
thedevelopmentofshortpitchcorrugation,andonceacertainseverityisreached,verticalmodes
becomedominant;and(3)inthecasesimulatedinthispaper,corrugationdoesnotgrowprobably
duetonotonlythedifferentresultingmainfrequenciesoftheverticalandlongitudinalcontactforces,
butalsotheinconsistencybetweenthefrequenciesoftheverticalandlongitudinalvibrationmodes
andtheresultingwear. Itisinferredthatinthecontinuousprocessofinitiationandgrowthofthe
corrugation,thereshouldbeaconsistencybetweenthem,andthiscouldbedonebythecontrolof
certaintrackparameters.
Keywords:shortpitchcorrugation;developmentmechanism;3D-FEdynamicvehicle-trackmodeling;
verticalandlongitudinaltrackvibrations;frictionalrollingcontact
1. Introduction
Railcorrugationsareperiodicdefectscommonlyobservedinalltypesofrailwaytracks.According
to the current understanding of their development mechanisms, namely the damage mechanisms
and wavelength-fixing mechanisms [1], rail corrugations can be classified into six groups: heavy
haul,lightrail,bootedsleepers,contactfatigue,ruttingandshortpitchcorrugation. Amongthese
groups,thedevelopmentmechanismsofshortpitchcorrugation(hereaftercorrugation)arenotfully
understood. Although corrugation was identified more than a century ago and despite extensive
researchefforts[1,2],aneffectivesolutionhasnotbeendevelopedtoavoidthecorrugationproblem,
whichremainsanenigmathathaspuzzledmanyresearchersandengineers[3,4]. Grindingappearsto
betheonlyeffectivecorrectivecountermeasure;however,itincreasesmaintenancecostsandreduces
theavailabilityoftherailwaynetwork. Thus,thedevelopmentmechanismsofthisphenomenonmust
beelucidated,andaneffectivesolutionforitscontrolatanearlystagemustbeidentified.
Appl.Sci.2017,7,807;doi:10.3390/app7080807 www.mdpi.com/journal/applsci
Appl.Sci.2017,7,807 2of23
Corrugation mainly occurs on straight tracks or gentle curves where contact does not occur
betweenthewheelflangeandrailgaugecorner(Figure1),anditusuallymanifestsasshinyripplesand
darkvalleys. Thetypicalwavelengthisintherangeof20–80mm,withamplitudesupto100µm[1].
Corrugationisoneofthemostprominentproblemsforrailwayinfrastructuremanagersbecauseit
increasesthevibrationsofthevehicle-tracksystemandresultsinhigherfrequencies(morethan500Hz)
ofthewheel-raildynamiccontactforces,whichleadstoaccelerateddegradationofvehicle-tracksystem
componentsandshortenedservicelife[5]. Inaddition,thenoisegeneratedbyvibrationsisanuisance
to residents living near railway lines [1,6]. Because of the high level of noise, corrugation is also
knownas“roaringrail”. Corrugationcanalsogeneraterollingcontactfatigue(RCF),suchassquats[7]
(Figure1c).
Figure1.ShortpitchcorrugationandtheresultingsquatsintheDutchrailwaynetwork.Wavelength
andperiodicityaredistinguished(wavelengthisthedistancebetweentwoadjacentshiningspotsof
ripples,andperiodicityreferstotheperiodicpatternthatcontainsmultiplewavelengthsofnon-uniform
amplitude of corrugation). (a) is at a gentle curve, and (b,c) are on straight tracks. (a) Uniform
corrugationwithawavelengthofapproximately30mm.Squatshavenotyetdeveloped,andaballasted
trackwithmono-blocksleepersandfasteningswithaW-shapedtensionclampisshown.Phototaken
near Assen, the Netherlands. (b) Non-uniform corrugation of a constant periodicity of a sleeper
span.Thecorrugationwavelengthvarieslargelywithinaperiod.Squatshavenotyetdeveloped,and
aballastedtrackwithduo-blocksleepersandfasteningswithDeenikclipsisshown.Phototakennear
Steenwijk,theNetherlands.(c)Non-uniformcorrugationwithaperiodicityshorterthanasleeperspan.
Thecorrugationwavelengthisapproximately30mm. Thesquatswerecausedbycorrugation,and
aballastedtrackwithduo-blocksleepersandfasteningswithDeenikclipsisshown.Phototakennear
Steenwijk,TheNetherlands.
The main damage mechanism of corrugation is commonly considered to be wear caused by
longitudinalwheelslip[8,9]. Plasticdeformationisanotherpossibledamagemechanism[1],andit
hasbeeninvestigatedbyanumericalapproach[10]andmetallurgicalanalyses[11]. Thewearand
Appl.Sci.2017,7,807 3of23
deformationunderlyingcorrugationaredifferential,i.e.,theyareselectiveprocessesthat“consistently”
occurmoreatcertainlocationsthanatadjacentlocations,whichresultsintheaccumulationandgrowth
of typical wave patterns [12]. In this case, “consistently” indicates that the wear and deformation
caused by one-wheel passage repeats the same wavelengths and phase angle of previous wheel
passages,whichresultsintheaccumulationofwearanddeformationsandtheinitiationofcorrugation
growth. Becausedamageoriginatesinthecontactpatch,thecontactmechanicswithinthecontact
patchmustbeexaminedtobetterunderstandthedamagemechanisms. Becausedynamicwheel-rail
contact is difficult to measure, a numerical analysis that can accurately simulate the vehicle-track
dynamicinteractionmustbedevelopedtoanalyzecorrugationundercontrolledconditions[13].
TheHertziantreatmentofthecontactproblemhasusuallybeenusedtosolvenormalcontact
problems[9],whereasKalker’stheories[14]aretypicallyemployedforthetangentialdirection[9].
TheHertziansolutionconsidersthatcontactsurfacesarefrictionlesssmoothhalf-spaces. However,for
wornordeformedprofiles,suchsurfaceapproximationsmaynotholdinthevicinityofthecontact
patch[15,16]. Nielsen[16]employedatwo-dimensional(2D)non-Hertziancontactmodelandfound
thattheshapeofthenormalstressdistributionwasnotellipticbecauseofthegeometricalasymmetry.
Correspondingly,thetangentialstressdistributionvariesalongdifferentpositionsofthecorrugation
andmayberesponsibleforthedevelopmentofcorrugation. Inaddition, theHertziantheoryand
Kalker’stheoryarebasedonstatics. However,becausethecontactpatchisontheorderof2cm,which
issimilartothecorrugationwavelength,non-steadystateprocessescausedbydynamicinteractions
mustbeconsidered[17,18].
Inadditiontolong-termdamagemechanisms,ashort-termdynamicprocessisbelievedtofix
thewavelengththroughstructuraldynamics[1,19]. Theinteractionbetweentheshort-andlong-term
mechanismsisrepresentedbyafeedbackloop[1]. Threephenomenamustbedefinedtounderstand
corrugation development: (1) structural dynamics excitation caused by vehicle-track interactions,
(2)loadingresponseatthewheel-railinterfaceinvolvingcontactmechanicsand(3)feedbackfrom
the contact and damage to the structural dynamics as determined by the direct coupling between
the contact mechanics and structural dynamics. In [9,20], a crucial understanding of the contact
phenomenainthecorrugationproblemisprovided. Themodelusedin[9,20]solvedthestructural
dynamicsandcontactmechanicsproblemsseparatelyindifferentmodelsorsteps.Inadditiontoamore
elaborated treatment of the contact mechanics and an accurate representation of the vehicle-track
interaction,weproposeacombinedmodelingapproachthatincludesthecouplingandinterplayof
boththestructuraldynamicsandcontactmechanicsproblems. Thissolutionisinspiredbythework
of[9]andthequestion“whyisroughnessgrowthpredictedbyasimplecontactmodelbutnotwhen
complexityisincreasedtoincludenon-Hertzianandnon-steadycontactconditions?”
The finite element method has been employed to investigate the development mechanism
underlyingsquatsinavehicle-tracksystem[7]. Thismodelingapproachprovidesagoodexplanation
forthedevelopmentofcorrugationinitiatedfromisolatedknownrailheadirregularities[21]. Because
corrugationfromisolatedknownrailheadirregularitiesisofashortpitchtype,thismethodcould
alsobevalidforinvestigatinggeneraltypesofshortpitchcorrugation. Corrugationthatoriginates
fromrailheadirregularitiesiscausedbyadynamicforceexcitedbyknownirregularities. Inthiscase,
the wavelength of the force is determined by the local track system, and the phase of the force is
determinedbythelocationoftheirregularities. Thus,clearmechanismsofwavelengthandphase
fixing are involved. The wavelength and phase of the force at a given irregularity are always the
same. Consequently,theresultingdifferentialwearand/orplasticdeformationisalwaysinphaseand
damageaccumulatesunderdifferentwheelpassages. Inthisway,corrugationcaninitiateandgrow.
Inthecaseofthegeneraltypeofshortpitchcorrugation,avisually-identifiableinitiationsource
isnotobserved. Althoughthewavelengthmightbefixedbythetrackstructure,e.g.,the“pin-pin”
resonance[22]orthestick-slipprocess[23],therandomnessofpassingwheelsandthetrackcanlead
tophasevariationsofthecontactforcesothatthetotaleffectofmanywheelpassagesmaycancelout
orsuppressthedifferentialwearanddeformation.
Appl.Sci.2017,7,807 4of23
Regardingthelattersituation,wemustidentifythecausesofthedynamicforcesthatresultin
differentialwearanddifferentialplasticdeformation,whichremaininphaseforthemanydifferent
passingwheelssuchthatthewearanddeformationaccumulateandcorrugationcaninitiateandgrow.
Althoughtheentiremechanismoftheinitiationanddevelopmentofshortpitchcorrugationwasnot
identifiedinthispaper,wepresentnewinsightsbasedonnumericalmodelingthatareconsistentwith
fieldobservationsandcancontributetoabetterunderstandingofthisenigma. Weexpectthatthenew
insightsprovidedherewilltriggermanynewresearcheffortsinthefieldthatwillultimatelyproduce
amorecompleteunderstandingoftheenigmaandafinalsolution.
In this paper, a 3D finite element (FE) approach is presented. This approach combines
the vehicle-track interaction model of [7] with the solution for transient frictional rolling of [24].
Thevehicle-trackstructuraldynamicsaredirectlycoupledtothewheel-railcontactmechanicsthrough
thecontinuumtreatmentofthewheelandrailinthestructure.Thesimulationofcontinuummechanics
hasbeenreportedtoplay“animportantroleinstructuralanalysis”[25]andallowstheinstantaneous
mutualinfluenceofthecontactmechanicsandthestructuralmechanicstobesimulated. Thisapproach
waspresentedin[26]toanalyzethephaserelationshipbetweenthegivencorrugationandtheresulting
periodicwear,andasimilarapproachwasemployedtostudycorrugationinahigh-speedrailway[27].
In this paper, the transient states of the rolling contact and wear are evaluated under a variety of
loadingconditionsinrelationtothedynamicforcesexcitedbyapassingwheeloverarailwithand
withoutcorrugation. Thedamagemechanismisassumedtobelimitedtowearbecausethefocusof
thispaperisontheconditionsthatmayleadtotheconsistentinitiationandgrowthofcorrugation.
To account for plasticity, the method of [28] can be readily incorporated, which will be our future
researchfocus.
2. Model
2.1. FEModel
Aschematicdiagramofthe3DFEvehicle-trackmodelisshowninFigure2.Themodelisbasedon
asymmetricalvehicle-tracksystemassumingastraighttrackinwhichlateralmovementisnegligible.
Thewheel, railandsleepersaremodeledwithsolidelements. Thenominalradiusofthewheelis
0.46 m, and the tread coning is 1/40. The rail is a standard International Union of Railway (UIC)
54profilewithaninclinationof1/40. Thecontactsurfaceofthewheelissmooth,whereasalengthof
corrugationisprescribedalongtherailsurface. Thesprungmass(theweightofthecarbodyandthe
bogie)islumpedintoamasselementM supportedbytheprimarysuspension,whichisrepresented
c
byspring-damperelements. Thesprungmassaboveahalfwheelsetis1/8ofthesprungdynamic
loadofawholevehicle,whichisapproximatelyaquarterofthesprungmasscarriedbyabogie. The
fasteningsystemandtheballastarealsomodeledasspring-damperelements. Thetrackparameters
aretakenfrom[29]asshowninTable1andrepresentthetypicalDutchrailwaysystem. IntheFE
model,thewheelandrailaremeshedwith8-nodesolidelements. Toachieveasolutionofsufficient
accuracyandanacceptablecomputationtime,onlythesizeoftheelementsinthesolutionzoneis
refined(0.8mm×0.8mminthelongitudinalandlateraldirections).Theelementsfarfromthesolution
zonearemeshedatanelementsizeupto7.5cm. Thesechoicesarebasedon[24],whichconcludedthat
thecontactmechanicssolutionwithanelementsizeof1.3mm×1.3mmissufficientlyaccuratewhen
theFEapproachusedhereisimplementedforengineeringapplications. Thetotalnumberofelements
in the model is 1,135,384; the number of nodes is 1,297,900; and the model length is 18 m. In [30],
atracklengthof10mwassufficientforproblemsofsimilarfrequencyandwavelength. Thedamage
mechanismstudiedinthispaperiswear;thus,thewheelandrailmaterialsareassumedtobeelastic.
ACoulombfrictionlawisemployedwithafrictioncoefficientf of0.6asin[31]. Intheliterature,the
C
frictioncoefficientofdrywheel-railcontactisreportedtobebetween0.4and0.65[32].
The solution process of the simulation includes two steps: an implicit analysis (using Ansys)
and an explicit analysis (using Ls-dyna). The implicit analysis is performed to identify the initial
Appl.Sci.2017,7,807 5of23
deformation in the equilibrium position of the vehicle-track system. Then, the wheel is set to roll
along the rail with a constant speed v = 38.9 m/s (corresponding to the typical Dutch passenger
trainspeedof140km/h). Anexplicitintegrationschemewithacentraldifferencemethodisthen
implementedtosolvethewheel-railfrictionalrollingcontactproblems. Thedisplacementsobtained
fromtheimplicitprocessareusedastheinitialstateoftheexplicitintegrationprocess. Ifthetime
step(4.67 × 10−8 sinthismodel)issmallerthanthecriticaltimestep(5 × 10−8 s)determinedby
theCourantcriterion[33], convergenceisguaranteed. Bykeepingthetimestepsufficientlysmall,
themodelcanincludeallnecessaryvibrationmodes. Intheexplicitanalysis,thefrictionalrollingis
modeledusingasurface-to-surfacealgorithmwiththepenaltymethoddescribedin[34].Becauseofthe
natureofexplicitintegration,theeffectoftransientrollingandthehigh-frequencydynamicbehavior
ofthevehicle-tracksystemexcitedbythemovingwheelareautomaticallyincludedinthesolution.
Figure2.Vehicle-trackfrictionalrollingmodelin3D.(a)Schematicdiagramofthemodel.(b)FEmodel
in3D.
Table1.Vehicleparametersandtrackparameters.
Parameters Values Parameters Values
Wheelload 116.8kN Young’smodulus 210GPa
Wheelandrail
Stiffness 1.15MN/m Poisson’sratio 0.3
Primarysuspension material
Damping 2.5kNs/m Density 7800kg/m3
Stiffness 1300MN/m Young’smodulus 38.4GPa
Railpad
Damping 45kNs/m Poisson’sratio 0.2
Sleeper
Stiffness 45MN/m Massdensity 2520kg/m3
Ballast
Damping 32kNs/m Spacing(L) 0.6m
AsshowninFigure2a,adistanceL isusedduringtheexplicitprocesstodiminishtheeffectof
1
vibrationexcitedbyimperfectinitialequilibriumbecauseofnumericalerrorsfromtheimplicitsolution.
Atotalof10wavesofcorrugationwithlengthL areintroducedafterL . Thetractioncoefficientµis
2 1
definedasfollows:
µ = F /F ≤ f (1)
L N C
whereF isthenormalcontactforceandF istheresultanttangential(creep)forceinthelongitudinal
N L
directioncausedbyanappliedtorque. Differenttractioncoefficientsproducedifferentadhesion-slip
states,aswellasdamages. Inthesubsequentanalysis,atractionloadcorrespondingto40%ofthe
staticnormalcontactforceisappliedviaatorqueabouttheaxisofthewheel. Thisproducesaµof
0.4,whichisusuallythemaximaltractioncoefficientofrollingstock[35,36]. Becauseofthedynamic
natureofvehicle-trackinteractions,especiallyinthepresenceofcorrugation,theactualcontactforces
arenotconstant. Therefore,theinstantaneoustractioncoefficientvarieswithtimeandspacealong
thetrack, andtheactualvalueofthetractioncoefficientdeviatesfromthestaticvalue. Regarding
Appl. Sci. 2017, 7, 807 6 of 22
Appl.Sci.2017,7,807 6of23
A few additional remarks on the model are warranted. Note that high friction and traction
coefficients are used. In [37,38], high traction was reported to facilitate corrugation development.
creepage,theapproachof[7,24]isusedinthispaper,whereaconstantdrivingtorqueisspecified. The
Additionally, regarding the traction coefficient, we assume that the damage mechanism is wear.
resultingcreepagefluctuateswiththecorrugation[27].
Hence, a large value is chosen to ensure a high longitudinal contact force and thus obtain high wear.
A few additional remarks on the model are warranted. Note that high friction and traction
A large traction coefficient indicates a large slip zone. A large slip zone aids in the visualization of
coefficients are used. In [37,38], high traction was reported to facilitate corrugation development.
changes in the contact patch caused by corrugation; thus, it is easier to visualize and compare the
Additionally,regardingthetractioncoefficient,weassumethatthedamagemechanismiswear. Hence,
contact solutions [24]. Finally, the model does not include brake disks or gears and will not represent
alargevalueischosentoensureahighlongitudinalcontactforceandthusobtainhighwear. Alarge
torsional vibrations. Torsional wheelset vibrations have been considered as a cause of corrugation at
tractioncoefficientindicatesalargeslipzone.Alargeslipzoneaidsinthevisualizationofchangesinthe
curves [1,39,40]; however, recent research [41] has shown that torsional wheelset vibrations are not
contactpatchcausedbycorrugation;thus,itiseasiertovisualizeandcomparethecontactsolutions[24].
correlated with corrugation development at curves. Torsional vibration is also considered in [42] for
Finally,themodeldoesnotincludebrakedisksorgearsandwillnotrepresenttorsionalvibrations.
corrugation related to vibration up to about 100 Hz. In this paper, the frequencies considered are up
Torsional wheelset vibrations have been considered as a cause of corrugation at curves [1,39,40];
to 2000 Hz.
however, recent research [41] has shown that torsional wheelset vibrations are not correlated with
corrugationdevelopmentatcurves.Torsionalvibrationisalsoconsideredin[42]forcorrugationrelated
2.2. Corrugation Model
tovibrationuptoabout100Hz. Inthispaper,thefrequenciesconsideredareupto2000Hz.
In this paper, corrugation with a sinusoidal profile is used as a test case. This choice is based on
2.2. CorrugationModel
field observations, such as those shown in Figure 1a. Moreover, our work is a continuation of
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fieAld soinbsuesroviadtaiol ncso,rsruucghataisonth owsiethsh ao wconnisntaFnigt uwreav1ae.leMngotrhe oivse ar,populriewd otrok tihsea croainlt isnuurafaticoen eoxfpprreesvsieodu sby
retsheea recqhuiantiownh: ich corrugation was assumed to have a sinusoidal profile [3,9,43–46]. A sinusoidal
corrugationwithaconstantwavelengthisappliedto2tπh(cid:1876)erailsurfaceexpressedbytheequation:
(cid:1878)(cid:4666)(cid:1876)(cid:4667)=Acos(cid:3436) (cid:3397)θ(cid:3440)−A (2)
(cid:18) λ (cid:19)
2πx
z(x) =Acos +θ −A (2)
where A is the amplitude of the corrugation, λ iλs the wavelength and θ is the phase. The second
term −A guarantees that the peak of the corrugation is not higher than the initial profile of the
whraeirle sAurifsacteh.e amplitudeofthecorrugation,λisthewavelengthandθisthephase. Thesecondterm
−AguTarhaen tweeasvethleantgththe pise a3k0 omfmth,e wcohrircuhg iast iaopnpisronxoimthaitgehlye retqhuaanl tthoe oinnieti aolf ptrhoefi lreecoofrtdheedr aciolrsruurgfaactieo.n
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foBryco mrrauignatateindinragi lθa rbeemtwoedeenle −dπ(/t2h eanpde aπk,- ctoo-nttraocutg sholduitsitoannsc ecains tbwei scteutdhieedam atp dliitfufedree)n.tB lyocmataiionntas iwniinthgin
θobnetew ceoemnp−leπte/ 2coarnrudgπa,ticoonn wtaactvesolelnugtitohn, si.eca., nthbee fsatlulidngie dedagted (iPff1e)r, etnhte ltorcoautgiohn (sPw2)i,t thhine roinsiengco emdgpele (tPe3)
coarnrudg tahtieo npewakav (ePle4n),g aths ,sih.eo.,wthne ifna lFliinggureed g3eb.( PF1ig),utrhee 3trco suhgohw(sP 2a) ,mthaegrniisfiinegd e3dDg eco(Pn3fi)gaunrdattihoen poefa tkhe
(Pc4o),rrausgsahtoiownn inin thFeig ruarile s3ubr.faFcieg.u re3cshowsamagnified3Dconfigurationofthecorrugationinthe
railsurface.
(a)

Figure3.Cont.
AAppplp.lS. Scic.i.2 2001177,,7 7,,8 80077 77o off2 322
(b)

(c)

Figure3.Modeledcorrugation:(a)fieldmeasurementofcorrugation(27–33mmbandpassfiltering);
Figure 3. Modeled corrugation: (a) field measurement of corrugation (27–33 mm bandpass filtering);
(b)schematicdiagramofthecorrugationand4positions(blueline:P1,redline:P2,greenline:P3and
(b) schematic diagram of the corrugation and 4 positions (blue line: P1, red line: P2, green line:
magentaline:P4);and(c)illustrationoftheappliedcorrugation(Corrugation2withP2at0.6m)inthe
P3 and magenta line: P4); and (c) illustration of the applied corrugation (Corrugation 2 with P2 at 0.6 m)
railsurfacewith5×magnification(only3completewavesareplotted).
in the rail surface with 5× magnification (only 3 complete waves are plotted).
2.3. ValidityoftheModel
2.3. Validity of the Model
TocharacterizethedynamicsinthefrequencyrangeofinterestasdiscussedintheIntroduction,
To characterize the dynamics in the frequency range of interest as discussed in the Introduction,
corrugation modeling should consider the following issues: static and dynamic contact problems
corrugation modeling should consider the following issues: static and dynamic contact problems
shouldbevalidated;structuraldynamicsshouldbeconsidered[47];andcouplingbetweenthecontact
should be validated; structural dynamics should be considered [47]; and coupling between the
problemandstructuraldynamicsshouldbevalidated.
contact problem and structural dynamics should be validated.
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Appl.Sci.2017,7,807 8of23
AswillbediscussedingreaterdetailinSection4.2onthepredictionofmajorfieldobservations,
thesimulatedwearinSection4.2isinagreementwiththefieldobservationsshowninFigure1b,c.
Here, the irregular distribution of wear along one sleeper span is clear, and the observations and
simulation are in agreement. In the paper, other agreements between the model results and field
observationsarediscussedingreaterdetaillaterinthetext. Forexample,thecontactsolutionsthat
indicatethatwearisthemoreprobabledamagemechanismatthecorrugationtrough[1,11,52]willbe
describedinSection3,andtheoccurrenceofsquatsinagreementwiththeobservationinFigure1c
willbeexplainedinSection4.2. Thesesummariescomprehensivelydescribeandanalyzehowthe
modelinthispaperwasvalidated.
For other track and vehicle field data, typical parameters of the Dutch railway are used [29].
Formodelingcorrugation,asinusoidalprofileisemployedbasedonfieldmeasurements(Figure3a)
andobservations(Figure1a).
3. ContactSolutionsatCorrugation
3.1. NormalContact
The FE approach defines whether a node is within the contact patch through the node force.
Anodeisincontactif:
|F | >0 (3)
n_N
where F is the nodal force in thedirection normalto the local surface. Because of the presence
n_N
of corrugation, the stress and slip distributions will vary along a corrugation wavelength. Those
differential distributions of the normal pressure, shear stress and slip are the direct factors in
corrugationdevelopment.
Figure 4a shows the changes in the contact patch size at P1, P2, P3 and P4 with increasing
amplitude A. Compared with A = 0 µm, an increase in size of up to 16% at P2 is observed in the
case of A = 20 µm, and the increase is approximately 4% at P3. At the other two positions, the
changesareminor. Themaximumcontactpatchsizeisapproximately200mm2 (A=20µm)atP2,
i.e.,thecorrugationtrough,whereasitisapproximately170mm2whentherailissmooth. Figure4b
displaysthemaximumcontactpressurealongthelongitudinalaxis(y=0mm)fordifferentcorrugation
amplitudesA.Asthecorrugationamplitudeincreasesfrom0–20µm,thecontactpressureatP2drops
significantlyto56%oftheoriginallevel,i.e.,from1320–745MPa. Thisdecreaseispartlybecauseof
the15%increaseinthecontactpatchsize(Figure4a). AtP1andP3,themaximumpressuredeclines
slightlyto78%oftheoriginallevel. Anincreaseinpressureof6%isobservedatP4forthecaseof
A=20µm.
Figure5showsatypicalprofileofthevariationofthemagnitudeofnormalforce,contactsize
andmaximumcontactpressurealongthelongitudinalaxis. Inthefigure,thetypicalnormalforceis
thelowestattroughP2andthehighestatpeakP4,withintermediatevaluesatpointsP1andP3.
Asthecorrugationamplitudeincreases,thenormalcontactforceatP2decreasesandtheareaof
thecontactpatchincreases;thus,thepressuredecreases. AtP4,thenormalcontactforceincreases,and
theareaofthecontactpatchdecreases;thus,thepressureincreases. AtP1andP3,thenormalcontact
force and the area of the contact patch are intermediate between cases P2 and P4. These patterns
arefurtherevidentinFigure4b,wherethemaximumcontactpressureofP1andP3forthedifferent
corrugationamplitudesdecreases,althoughitdoesnotdecreaseasstronglyasinthecaseofP2.
Appl.Sci.2017,7,807 9of23
Figure4.Influenceofcorrugationamplitudeonthesizeofthecontactpatchandthemaximumcontact
pressurealongthelongitudinalaxisy=0mm.(a)Contactsize;(b)contactpressure.
Figure5.Magnitudeofthenormalforce,contactsizeandmaximumpressurealongthelongitudinal
axisat(y=0mm)atthefouranalyzedpositionsP1,P2,P3andP4(A=20µm).
Figure6showstheprojectionsofcontactpressureandshearstressontothexOyplane,theshape
ofthecontactpatchunderthesmoothconditionandtheevolutionwithinonecorrugationwavelength
fromP1toP4whenA=20µm. MostshapesarenotellipticandthusdifferfromtheHertziansolution.
Because of the downhill slope of the corrugation at P1, a large drop in pressure is observed at the
leadingedgeofthecontactpatchandviceversaatP3.Thus,thepressuredistributionsatP1andP3take
theshapesofa“waningcrescentmoon”and“waxingcrescentmoon”,respectively. AtP2,themaximal
pressureshiftstothefieldandgaugesides,therebydecreasingthecontactpressureinthecontactpatch
centerto600–800MPa. ThepressuredistributionatP4isclosetothesolutionunderthesmoothrail
exceptforaslightshrinkageofthecontactpatchattheleadingandtrailingedgecenters.Themaximum
pressureisslightlyhigherthanthatattheotherlocationsasshowninthepressurescalingcolorbar.
As also shown in Figure 5, the pressure variation within a corrugation wavelength along the
longitudinalaxisy=0mmisapproximatelyinphasewiththecorrugation. Furthermore,theshapesof
thecontactpatchdifferfromthoseunderthesmoothrailcondition. Partofthecontactpatchisnot
presentintheleadingareaatP1,inthetrailingareaatP3andinboththeleadingandtrailingareasat